Problem 39 - College Algebra with Modeling and Visualization

Question

Use transformations to explain how the graph of \(f\) can be found by using the graph of \(y=x^{2}\) \(y=\sqrt{x},\) or \(y=|x| .\) You do not need to graph \(y=f(x)\). \(f(x)=-\sqrt{x+5}\)

Step-by-Step Solution

Answer

Shift left 5 units and reflect over the x-axis from \(y = \sqrt{x}\).

Step 1: Identify Base Function

The given function is \(f(x) = -\sqrt{x+5}\). The base function for this transformation is \(y = \sqrt{x}\), which is the square root function.

Step 2: Horizontal Shift

The term \(x + 5\) inside the square root indicates a horizontal shift. The graph of the base function \(y = \sqrt{x}\) is shifted 5 units to the left. This is because the general form \(y = \sqrt{x + c}\) shifts the graph left by \(c\) units when \(c\) is positive.

Step 3: Reflection about the X-axis

The negative sign in front of the square root, \(-\sqrt{x+5}\), results in a reflection of the graph over the x-axis. This changes the direction of the graph from upward to downward.

Key Concepts

Base Function

The concept of a base function makes understanding graph transformations much simpler. A base function acts as our starting point or reference. For example, in the given function \(f(x) = -\sqrt{x+5}\), the base function is \(y = \sqrt{x}\), which is known as the square root function. The square root function has a unique shape, starting from the origin and curving gently upward. By identifying the base function, we are better equipped to visualize and apply transformations that shift or alter its graph. Always begin with knowing the characteristics of the base function you're dealing with, as this provides clarity on how transformations like shifts and reflections will change the graph.

Horizontal Shift

Horizontal shifts occur when a constant is added or subtracted inside the function's argument (the part that goes inside the square root, parenthesis, etc.). In the case of \(f(x) = -\sqrt{x+5}\), the \(+5\) indicates a horizontal shift. Specifically, the rule is that \(y = \sqrt{x+c}\) results in a shift to the left by \(c\) units if \(c\) is positive and to the right if \(c\) is negative. In this case, \(+5\) moves the graph 5 units left. Visualizing it: each point on the graph of \(y = \sqrt{x}\) is moved 5 units to the left to create the new positions. This does not change the shape of the graph, only the horizontal placement.

Reflection

Reflection is a transformation that "flips" the graph over a given axis. In our example, the function \(f(x) = -\sqrt{x+5}\) contains a negative sign in front of the square root. This negative sign results in a reflection over the x-axis. But what does that mean? For the square root function, normally the graph "grows" upwards from the origin. However, when the graph is reflected over the x-axis, each point on the graph is mirrored below the x-axis, effectively inverting the graph. The upward facing curve of \(y = \sqrt{x}\) turns downward, but maintains its general shape. Reflection is straightforward when you think of it as simply flipping the graph to the opposite side of the x-axis.